The bound (REF ) is a step towards making the analysis of
Ref. [1]} quantitatively precise. At least for the
\(R^3\)  correction to the graviton three-point coupling which is
considered here, it translates the problem to bounding the coefficient
\(\beta _{R^4}^+\)  in terms of \(G / m_{\rm gap}^2\) .Recall
from Eq. (REF ) that \(\left( {\kappa /2} \right)^2 = 8 \pi G\) . This problem was
beautifully solved recently in Ref. [2]} for
\(D=10\)  maximal supergravity in a perturbative setting similar to
ours, and it was addressed nonperturbatively in
Ref. [3]}. The method used in
Ref. [2]} is not directly applicable in four
dimensions due to the IR divergences, but at least in \(D\ge 5\)  where
the \(2 \rightarrow 2\)  amplitude is nonperturbatively well-defined it is
natural to expect that one will be able to get a bound on
\(\beta _{R^4}^+\)  in terms of \( {G / m_{\rm gap}^2}\) . It would be
very interesting to demonstrate it explicitly.
